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Claim. Let $f(N)$ be the largest size of a set $A\subseteq\{1,\dots,N\}$ with $a+a'$ squarefree for all $a,a'\in A$, the case $a=a'$ included. P. Erdős and A. Sárközy, On divisibility properties of integers of the form $a+a'$, Acta Math. Hungar. 50 (1987), no. 1--2, 117--122, prove two bounds. Theorem 1: for $N>N_0$ there is such a set with $|A|>\frac1{248}\log N$. Theorem 2: for $N>N_1$ every such set has $|A|<3N^{3/4}\log N$. So

$$\frac{\log N}{248}<f(N)<3N^{3/4}\log N$$

for all large $N$. The authors expect the lower bound to be nearer the truth and conjecture that the upper bound can be replaced by $N^\varepsilon$ for every $\varepsilon>0$, perhaps even by $(\log N)^c$; these are the two questions of Problem 1109. The source card is Erdős and Sárközy 1987. The DOI record gives the issue as March 1987 and no day, so the page is named by the first day of that month.

Covers. The first estimates of $f(N)$: $\log N\ll f(N)\ll N^{3/4}\log N$. Neither question of the problem is answered; both bounds are improved by Konyagin 2004.

Depends on. No page of this wiki.

Acceptance. Refereed: Acta Math. Hungar. 50 (1987), no. 1--2, 117--122. The site labels the problem OPEN, so its commentary crediting the bounds is not reviewed evidence. The proof is not checked in this corpus.